Probability and Statistics for STEM

✍ Scribed by Emmanuel N. Barron

• Publisher: Springer Nature
• Year: 2024
• Tongue: English
• Leaves: 298
• Edition: 2
• Category: Library

✦ Synopsis

This new edition presents the essential topics in probability and statistics from a rigorous standpoint. Any discipline involving randomness, including medicine, engineering, and any area of scientific research, must have a way of analyzing or even predicting the outcomes of an experiment. The authors focus on the tools for doing so in a thorough, yet introductory way. After providing an overview of the basics of probability, the authors cover essential topics such as confidence intervals, hypothesis testing, and linear regression. These subjects are presented in a one semester format, suitable for engineers, scientists, and STEM students with a solid understanding of calculus. There are problems and exercises included in each chapter allowing readers to practice the applications of the concepts.

✦ Table of Contents

Preface to the Second Edition
Preface to the First Edition
Contents
About the Authors
1 Probability
1.1 The Basics
1.1.1 Equiprobable Sample Spaces
1.2 Conditional Probability
1.2.1 Independence
1.2.2 Bayes' Theorem
1.3 Appendix: Counting Techniques
1.3.1 Multiplication Principle
1.3.2 Permutations
1.3.3 Combinations
1.4 Problems
2 Random Variables
2.1 The Basics
2.1.1 Discrete RVs and PMFs
2.1.2 Cumulative Distribution Functions
2.1.3 Continuous RVs and PDFs
2.2 Important Discrete Distributions
2.2.1 Discrete Uniform RVs
2.2.2 Bernoulli RVs
2.2.3 Binomial RVs
2.2.4 Geometric RVs
2.2.5 Negative Binomial RVs
2.2.6 Poisson RVs
2.2.7 Hypergeometric RVs
2.2.8 Multinomial RVs
2.2.9 Simulating Discrete RVs Using a Box Model
2.3 Important Continuous Distributions
2.3.1 Uniform RVs
2.3.2 Exponential RVs
2.3.3 Normal RVs
2.4 Expectations, Variances, Medians, and Percentiles
2.4.1 Expectation
2.4.2 Variance
2.4.3 Medians and Percentiles
2.5 Moment-Generating Functions
2.6 Joint Distributions
2.6.1 Two Discrete RVs
2.6.2 Two Continuous RVs
2.6.3 Expected Values
2.7 Independent RVs
2.7.1 Conditional Distributions
2.7.2 An Application of Conditional Distributions: Bayesian Analysis
2.7.3 Covariance and Correlation
2.7.4 The General Central Limit Theorem
2.8 Chebychev's Inequality and the Weak Law of Large Numbers
2.9 Other Distributions Important in Statistics
2.9.1 Chi Squared Distribution
2.9.2 Student's t Distribution
2.9.3 Fisher–Snedecor F Distribution
2.10 TI-8x Commands
2.11 Problems
3 Distributions of Sample Mean and Sample SD
3.1 The Statistics overlineX, S2 and widetildeX of a Random Sample
3.2 Normal Populations
3.2.1 XsimN(µ,σ), σ Known
3.2.2 XsimN(µ,σ), σ Unknown
3.2.3 The Population X is not Normal but has Known Mean and Variance
3.2.4 The Population is Bernoulli, p is Known
3.2.5 The Population is Bernoulli, p is Unknown
3.3 Sampling Distributions of Differences of Two Samples
3.4 The Median: Order Statistics and the Central Limit Theorem
3.4.1 Continuous RVs and Order Statistics
3.4.2 Discrete RVs
3.4.3 Order Statistics and Sample Percentiles
3.5 Problems
4 Confidence and Prediction Intervals
4.1 Confidence Intervals for a Single Sample
4.1.1 Controlling the Error of an Estimate Using Confidence Intervals
4.1.2 Pivotal Quantities
4.1.3 Confidence Intervals for the Mean and Variance of a Normal Distribution
4.1.4 Confidence Intervals for a Proportion
4.1.5 One-Sided Confidence Intervals
4.2 Confidence Intervals for Two Samples
4.2.1 Difference of Two Normal Means
4.2.2 Confidence Interval for the Ratio of Variances
4.2.3 Difference of Two Binomial Proportions
4.2.4 Paired Samples
4.3 Prediction Intervals
4.4 Problems
5 Hypothesis Testing
5.1 A Motivating Example
5.2 The Basics of Hypothesis Testing
5.3 Hypotheses Tests for One Parameter
5.3.1 Hypotheses Tests for the Normal Parameters, Critical Value Approach
5.3.2 The P-Value Approach to Hypothesis Testing
5.3.3 Test of Hypotheses for Proportions
5.4 Hypotheses Tests for Two Populations
5.4.1 Test of Hypotheses for Two Proportions
5.5 Power of Tests of Hypotheses
5.5.1 Factors Affecting Power of a Test of Hypotheses
5.5.2 Power of One-Sided Tests
5.6 More Tests of Hypotheses
5.6.1 Chi-Squared Statistic and Goodness-of-Fit Tests
5.6.2 Contingency Tables and Tests for Independence
5.6.3 Analysis of Variance
5.7 Multiple Testing Problem and ANOVA
5.7.1 Bonferroni and Šidák Corrections
5.7.2 Tukey's Simultaneous Confidence Intervals
5.8 Problems
5.9 Summary Tables
6 Linear Regression
6.1 Introduction and Scatter Plots
6.2 Introduction to Regression
6.2.1 The Linear Model with Observed X
6.2.2 Estimating the Slope and Intercept from Data
6.2.3 Errors of the Regression
6.3 The Distributions of and
6.4 Confidence Intervals for Slope and Intercept & Hypothesis Tests
6.4.1 Hypothesis Tests for Slope and Intercept
6.4.2 ANOVA for Linear Regression
6.4.3 Confidence and Prediction Bands
6.4.4 Hypothesis Test for the Correlation Coefficient
6.5 Problems
7 Appendix: Answers to Problems
7.1 Answers to Chap. 1 Problems
7.2 Answers to Chap. 2 Problems
7.3 Answers to Chap. 3 Problems
7.4 Answers to Chap. 4 Problems
7.5 Answers to Chap. 5 Problems
7.6 Answers to Chap. 6 Problems
Index
Index